Harmonic Oscillator with fermions

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SUMMARY

The discussion focuses on the energy change of a system of three fermions (s=1/2) in the ground state of a harmonic oscillator, described by the Hamiltonian H = (1/2m)(P1)^2 + (1/2)m(w^2)(x1)^2 + (1/2m)(P2)^2 + (1/2)m(w^2)(x2)^2 + (1/2m)(P3)^2 + (1/2)m(w^2)(x3)^2. A perturbation W = g*m*(w^2/h)*(S1z*x2*x3 + S2z*x3*x1 + S3z*x1*x2) is introduced, affecting the ground state's energy. The discussion aims to analyze how this perturbation influences the energy levels of the fermionic system.

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milesAhead
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Hello...
We have 3 fermions (s=1/2) at the ground state of a harmonic oscillator moving over the x-axis with a the classic hamiltonian for a three particle oscillator :
H =(1/2m)*(P1)^2 +((1/2)*m(w^2)((x1)^2)) +(1/2m)*(P2)^2 +((1/2)*m(w^2)((x2)^2)) +(1/2m)*(P3)^2 +((1/2)*m(w^2)((x3)^2))

we have a pertubation between fermions that is:
W=g*m*((w^2)/h)*(S1z*x2*x3+S2z*x3*x1+S3z*x1*x2).

And it is asked the chande of the ground state's energy due to the pertubation.
 
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With S1z,S2z,S3z the spin (z) of each fermion
 

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